Optimal. Leaf size=51 \[ -\frac {\tan (c+d x)}{d \sqrt {b \tan ^4(c+d x)}}-\frac {x \tan ^2(c+d x)}{\sqrt {b \tan ^4(c+d x)}} \]
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Rubi [A]
time = 0.02, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3739, 3554, 8}
\begin {gather*} -\frac {\tan (c+d x)}{d \sqrt {b \tan ^4(c+d x)}}-\frac {x \tan ^2(c+d x)}{\sqrt {b \tan ^4(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3554
Rule 3739
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {b \tan ^4(c+d x)}} \, dx &=\frac {\tan ^2(c+d x) \int \cot ^2(c+d x) \, dx}{\sqrt {b \tan ^4(c+d x)}}\\ &=-\frac {\tan (c+d x)}{d \sqrt {b \tan ^4(c+d x)}}-\frac {\tan ^2(c+d x) \int 1 \, dx}{\sqrt {b \tan ^4(c+d x)}}\\ &=-\frac {\tan (c+d x)}{d \sqrt {b \tan ^4(c+d x)}}-\frac {x \tan ^2(c+d x)}{\sqrt {b \tan ^4(c+d x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.06, size = 43, normalized size = 0.84 \begin {gather*} -\frac {\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\tan ^2(c+d x)\right ) \tan (c+d x)}{d \sqrt {b \tan ^4(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 40, normalized size = 0.78
method | result | size |
derivativedivides | \(-\frac {\tan \left (d x +c \right ) \left (\arctan \left (\tan \left (d x +c \right )\right ) \tan \left (d x +c \right )+1\right )}{d \sqrt {b \left (\tan ^{4}\left (d x +c \right )\right )}}\) | \(40\) |
default | \(-\frac {\tan \left (d x +c \right ) \left (\arctan \left (\tan \left (d x +c \right )\right ) \tan \left (d x +c \right )+1\right )}{d \sqrt {b \left (\tan ^{4}\left (d x +c \right )\right )}}\) | \(40\) |
risch | \(\frac {\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} x}{\sqrt {\frac {b \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}}\, \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {2 i \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{\sqrt {\frac {b \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}}\, \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2} d}\) | \(120\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 27, normalized size = 0.53 \begin {gather*} -\frac {\frac {d x + c}{\sqrt {b}} + \frac {1}{\sqrt {b} \tan \left (d x + c\right )}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 39, normalized size = 0.76 \begin {gather*} -\frac {\sqrt {b \tan \left (d x + c\right )^{4}} {\left (d x \tan \left (d x + c\right ) + 1\right )}}{b d \tan \left (d x + c\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {b \tan ^{4}{\left (c + d x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.54, size = 45, normalized size = 0.88 \begin {gather*} -\frac {\frac {2 \, {\left (d x + c\right )}}{\sqrt {b}} - \frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {b}} + \frac {1}{\sqrt {b} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sqrt {b\,{\mathrm {tan}\left (c+d\,x\right )}^4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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